Dynamical Analyses on Beta Oscillations in a STN-GPE-GPI Model of Parkinson’s Disease

<div>Typical time series of the firing rate of GPi for some parameters: (a<sub>1</sub>)–(a<sub>3</sub>) <span class="nowrap"><svg height="12.2715pt" id="M94" style="vertical-align:-3.30938pt" version="1.1" viewbox="-0.0498162 -8.96212 58.7103 12.2715" width="58.7103pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M962 650H739L732 622L760 619C812 613 819 604 798 552C760 457 671 267 606 131H604L511 638H480L237 131H233L183 554C177 607 183 614 226 619L248 622L257 650H24L17 622C88 615 92 611 103 524L173 -11H203L450 491H453L543 -11H575L839 529C882 609 886 613 953 622L962 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.777,3.132)"><path d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,17.284,3.132)"><path d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.242,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,37.779,0)"><path d="M620 675H597C578 656 570 650 541 650H144C112 650 104 653 94 675H72C59 618 42 552 23 493L53 491C71 534 88 564 105 585C124 608 144 615 238 615H290L197 121C182 40 174 34 88 28L82 0H361L367 28C275 34 266 38 281 121L374 615H441C522 615 543 608 553 583C562 560 566 531 565 493L597 494C603 551 612 629 620 675Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,44.955,3.132)"><path d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,51.461,3.132)"><path d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g></svg>,</span> (b<sub>1</sub>)–(b<sub>3</sub>) <span class="nowrap"><svg height="12.2715pt" id="M95" style="vertical-align:-3.30938pt" version="1.1" viewbox="-0.0498162 -8.96212 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37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,21.597,3.132)"><path d="M303 0V28C221 34 213 39 213 125V525C213 610 221 616 303 622V650H38V622C120 616 128 610 128 525V125C128 40 120 34 38 28V0H303Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.156,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,38.692,0)"><path d="M620 675H597C578 656 570 650 541 650H144C112 650 104 653 94 675H72C59 618 42 552 23 493L53 491C71 534 88 564 105 585C124 608 144 615 238 615H290L197 121C182 40 174 34 88 28L82 0H361L367 28C275 34 266 38 281 121L374 615H441C522 615 543 608 553 583C562 560 566 531 565 493L597 494C603 551 612 629 620 675Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,45.868,3.132)"><path d="M409 504C401 567 396 607 392 642C354 654 312 665 266 665C137 665 60 583 60 487C60 374 161 325 225 290C300 250 355 215 355 141C355 68 311 21 235 21C131 21 86 122 71 183L41 176C48 128 61 42 68 21C78 16 93 8 118 0C142 -7 175 -15 216 -15C349 -15 438 69 438 174C438 287 344 333 265 374C186 414 138 449 138 522C138 576 172 631 249 631C336 631 363 562 380 499L409 504Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,50.182,3.132)"><path d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,56.688,3.132)"><path d="M303 0V28C221 34 213 39 213 125V525C213 610 221 616 303 622V650H38V622C120 616 128 610 128 525V125C128 40 120 34 38 28V0H303Z"></path></g></svg>,</span> and (c<sub>1</sub>)–(c<sub>3</sub>) <span class="nowrap"><svg height="11.9453pt" id="M96" style="vertical-align:-3.309401pt" version="1.1" viewbox="-0.0498162 -8.6359 66.225 11.9453" width="66.225pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M962 650H739L732 622L760 619C812 613 819 604 798 552C760 457 671 267 606 131H604L511 638H480L237 131H233L183 554C177 607 183 614 226 619L248 622L257 650H24L17 622C88 615 92 611 103 524L173 -11H203L450 491H453L543 -11H575L839 529C882 609 886 613 953 622L962 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.777,3.132)"><path d="M409 504C401 567 396 607 392 642C354 654 312 665 266 665C137 665 60 583 60 487C60 374 161 325 225 290C300 250 355 215 355 141C355 68 311 21 235 21C131 21 86 122 71 183L41 176C48 128 61 42 68 21C78 16 93 8 118 0C142 -7 175 -15 216 -15C349 -15 438 69 438 174C438 287 344 333 265 374C186 414 138 449 138 522C138 576 172 631 249 631C336 631 363 562 380 499L409 504Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,15.09,3.132)"><path d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,21.597,3.132)"><path d="M517 162C503 123 484 88 467 68C445 42 417 34 341 34C291 34 256 34 237 47C219 59 213 81 213 128V317H308C395 317 402 311 415 240H444V431H415C403 364 398 356 307 356H213V584C213 613 215 616 246 616H322C394 616 421 609 435 587C448 566 458 544 467 502L496 506C493 557 488 625 488 650H42V622C120 616 128 612 128 523V125C128 43 120 35 29 28V0H511C520 31 540 125 546 158L517 162Z"></path></g><g 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Complexity

fig6

Figure 6

Figure 6: Dynamical Analyses on Beta Oscillations in a STN-GPE-GPI Model of Parkinson’s Disease 